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Double coverings of curves and non-Weierstrass semigroups. (English) Zbl 1270.14014
Let $$X$$ be a (non-singular, projective, irreducible, algebraic) curve defined over an algebraically closed field of characteristic zero, and let $$P\in X$$. The Weierstrass semigroup $$H(P)$$ of $$X$$ at $$P$$ is the set of poles of regular function on $$X\setminus\{P\}$$. Thus $$H(P)$$ is indeed a subsemigroup of the additive semigroup $$\mathbb N_0$$ such that $$\#(\mathbb N_0\setminus H(P)$$ equals the genus of $$X$$ (The Weierstrass gap theorem).
Let $$H$$ be a subsemigroup of $$(\mathbb N_0,+)$$ which is named numerical provided that $$G(H):=\mathbb N_0\setminus H)$$ is finite; the genus of $$H$$ is $$\#G(H)$$. The subject matter addressed in the paper under review is related to the following question posed by Hurwitz around 1892 [A. Hurwitz, Math. Ann. XLI, 403–442 (1893; JFM 24.0380.02)]: Is any numerical semigroup $$H$$ equal to the Weierstrass semigroup at some point of a curve? If this is so, $$H$$ is called Weierstrass. The answer to this question is in general negative as R.-O. Buchweitz pointed out around 1980 [Lect. Notes Math. 777, 205–220 (1980; Zbl 0428.32016)]. He observed the following by considering elements of $$G(H)$$. For an integer $$m\geq 2$$, let $$G_m(H)$$ be the set of all sums of $$m$$ elements of $$G(H)$$. Thus if $$H$$ of genus $$g$$ were Weierstrass, then $$\#G_m(H)\leq (2m-1)(g-1)\, (*)$$, the dimension of a $$m$$-pluricanonical divisor of a curve of genus $$g$$. In fact, Buchweitz constructed non-Weierstrass semigroups by contradicting condition $$(*)$$; see also [J. Komeda, Semigroup Forum 57, No. 2, 157–185 (1998; Zbl 0922.14022)]. The least genus of Buchweitz’s examples is $$g=16$$. Numerical semigroups of genus at most eight are always Weierstrass; see [J. Komeda and A. Ohbuchi, Bull. Braz. Math. Soc. (N.S.) 39, No. 1, 109–121 (2008; Zbl 1133.14307)] and the references therein. Let $$\ell_g=\ell_g(H)$$ be the biggest element of $$G(H)$$. By the semigroup property, $$\ell_g\leq 2g-1$$. If $$\ell_g\geq 2g-4$$, then condition $$(*)$$ is always true for any numerical semigroup; cf. [G. Oliveira, Semigroup Forum 69, No. 3, 423–430 (2004; Zbl 1076.20052)] and the references therein. However, non-Weierstrass numerical semigroups with $$\ell_g\geq 2g-4$$ do exist: G. Oliveira and K.-O. Stöhr [Geom. Dedicata 67, No. 1, 45–63 (1997; Zbl 0904.14018)], F. Torres [Commun. Algebra 23, No. 11, 4211–4228 (1995; Zbl 0842.14023)]; see also N. Medeiros [J. Pure Appl. Algebra 170, No. 2–3, 267–285 (2002; Zbl 1039.14015)]. The basic tool in constructing such non-Weierstrass semigroups is the use of certain covering of curves and Buchweitz’s examples as a building block (Stöhr).
On the other hand, let $$m(H)$$ be the first positive element of a numerical semigroup $$H$$. If $$m(H)\leq 5$$, $$H$$ is always Weierstrass; see [J. Komeda, Manuscripta. Math. 76, No. 2, 193–211 (1992; Zbl 0770.30038)] and the references therein. There exist non-Weierstrass semigroups $$H$$ whenever $$m(H)\geq 13$$ (e.g. Buchweitz, loc. cit.). In the article under review, the author constructs examples of non-Weierstrass semigroups $$H$$ with $$m(H)=8$$ and with $$m(H)=12$$. To explain his method, for a numerical semigroup $$\tilde H$$ let us consider the associated numerical semigroup $$d_2(\tilde H):=\{h/2: \text{}h$$ is even

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H30 Coverings of curves, fundamental group 20M14 Commutative semigroups
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##### References:
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